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Since no datum point in the geodetic network can be accepted as stable until the analysis is performed, the network must be treated as a free network. In means that the network in itself does not contain enough information to be located in space. Examples are a leveling network without elevation information of any point, or a horizontal trilateration and combined net network without the known coordinates of any point and any know azimuth between a pair of points. Therefore, free networks can be freely translated or rotated or scaled in space, and can be considered as suffering from datum defect.Consider the linearized parametric adjustment model of a free network as with where l is n vector of observations, v is the n- vector of residuals, is the vector of the corrections to the approximate coordinates of the points to be determined, A is the configuration matrix, is the a priori variance factor, and QCˆdA vl=+A QA1T−l20QσCˆdN20σl is the cofactor matrix of the observations. The least squares criterion leads to the normal equations , where . Due to datum defects in the network, the coefficient matrix, N, of the normal equations is singular, i.e., det(N)=0. In the set of generalized inverses obtained by (9), subsets of special generalized inverses may be defined by introducing additional conditions. In the following computation upon pseudoinverse of matrix N is introduced. |
